Chromatic vs natural scale

chromatic_vs_natural.py

#!/usr/bin/env python
# Copyright (c) 2011 by Sergey Lyubka
# All rights reserved
#
# This program calculates note frequencies for chromatic and natural scale.

# Chromatic scale is built by splitting an octave (2x frequency bump) onto
# 12 equal intervals (semitones).
semitones = ['%.5f' % pow(pow(2, x), 1 / 12.0) for x in range(13)]
tones = [0, 2, 4, 5, 7, 9, 11]  # Major scale
print 'Frequencies of chromatic notes: (all octaves are the same)'
print 'These are relative frequencies of do, re, mi, fa, sol, la, si'
print [semitones[i] for i in tones]

# The natural (quint) scale is built by the recurring algorithm:
# 1. take the base note
# 2. build a quint interval up, by multiplying the frequency to 3/2. Quint
#    interval (3/2) is the most harmonious interval to the human ear after
#    the octave (2/1). Harmonius means that frequencies relate as integers:
#    2/1, 3/2, 4/3, 5/4, etc
# 3. if resulting note is > 2 (next octave), divide it by 2 to bring it back
#    to our octave.
#
# Note that quints can be built down, as well as up. In this case, frequencies
# will be different. That's why in harmonic scale, C# != Db, these have
# different frequencies! Most musicians cannot explain the difference.
#
# In chromatic scale, these are the same notes, cause the octave is
# split in 12 equal intervals. Almost all modern instruments use the same chromatic
# scale, meaning that modern music is inherently not harmonious! The difference is
# very hard to spot. I am unable to do it by ear, only by playing two notes of
# chromatic and harmonic scale together, the beating is obvious.
def calc_harmonic_quint_up(n):
  tone = pow(3, n)
  while tone > 2:
    tone /= 2.0
  return tone

print
print 'Frequencies of natural notes (first 21 octaves):'
for octave in range(21):
  notes = [calc_harmonic_quint_up(note + 8 * octave) for note in range(7)]
  print ['%.5f' % x for x in sorted(notes)]

print
print 'Note the difference in frequencies!'
print 'Our music is built on chromatic scale, and is dissonant by design.'

Script output:

Frequencies of chromatic notes: (all octaves are the same)
['1.00000', '1.12246', '1.25992', '1.33484', '1.49831', '1.68179', '1.88775']

Frequencies of natural notes (first 21 octaves):
['1.00000', '1.12500', '1.26562', '1.42383', '1.50000', '1.68750', '1.89844']
['1.01364', '1.14035', '1.20135', '1.35152', '1.52046', '1.60181', '1.80203']
['1.08244', '1.21775', '1.28289', '1.44325', '1.62366', '1.82662', '1.92434']
['1.02747', '1.15591', '1.30040', '1.46294', '1.54121', '1.73386', '1.95059']
['1.04149', '1.17168', '1.23436', '1.38865', '1.56224', '1.64581', '1.85154']
['1.11218', '1.25120', '1.31814', '1.48290', '1.66827', '1.87680', '1.97721']
['1.00209', '1.05570', '1.18766', '1.33612', '1.50314', '1.58355', '1.78149']
['1.07010', '1.20387', '1.26827', '1.42680', '1.60515', '1.69103', '1.90241']
['1.01576', '1.14273', '1.28557', '1.35435', '1.52364', '1.71410', '1.92836']
['1.02962', '1.08470', '1.22029', '1.37283', '1.54443', '1.62705', '1.83044']
['1.09950', '1.23694', '1.30311', '1.46600', '1.64925', '1.73748', '1.95467']
['1.04367', '1.17413', '1.32089', '1.39156', '1.56550', '1.76119', '1.98134']
['1.05791', '1.11450', '1.25382', '1.41054', '1.58686', '1.67175', '1.88072']
['1.00418', '1.12971', '1.27092', '1.33891', '1.50628', '1.69456', '1.78522']
['1.01789', '1.07234', '1.20638', '1.35718', '1.42979', '1.60851', '1.80957']
['1.08697', '1.14512', '1.28826', '1.44929', '1.63046', '1.71768', '1.93239']
['1.03177', '1.16074', '1.30584', '1.37570', '1.54766', '1.74112', '1.83426']
['1.04585', '1.10180', '1.23953', '1.39447', '1.46907', '1.65270', '1.85929']
['1.11683', '1.17658', '1.32365', '1.48911', '1.67525', '1.76487', '1.98548']
['1.06012', '1.19263', '1.34171', '1.41349', '1.59018', '1.78895', '1.88465']
['1.07458', '1.13207', '1.27358', '1.43278', '1.50943', '1.69810', '1.91037']

Note the difference in frequencies!
Our music is built on chromatic scale, and is dissonant by design.